Many jurisdictions have the concept of emergency or exigent circumstances such that they allow the police to enter premises in scenarios like the one in the question, without having to comply with the rules that pertain to non-emergency or non-exigent circumstances (such as having to obtain a warrant beforehand).

Generally speaking, the less urgent or serious the apparent circumstances, the more the police must do to enter; conversely, the more urgent or serious the circumstances, the less the police must do to enter. Some jurisdictions have frameworks that are essentially like the following, from less urgent/serious to more urgent/serious:

apply for a warrant from a magistrate or judge -> seek authorisation from an officer of particular rank or higher -> seek permission from the occupier -> no need to seek permission

england-and-wales

If the police "apprehend serious bodily injury" (Syed v Director of Public Prosecutions [2010] EWHC 81), they can enter the home and they can do so without a warrant. This is provided for at section 17(1)(e) Police and Criminal Evidence Act 1986:

17 Entry for purpose of arrest etc. (1) Subject to the following provisions of this section, and without prejudice to any other enactment, a constable may enter and search any premises for the purpose— ... (e) of saving life or limb or preventing serious damage to property.

At common law, the police also have a power to enter private property without a warrant to stop or prevent a 'breach of the peace'. According to Friswell v Chief Constable of Essex [2004] EWHC 3009 (QB):

(1) The purpose of the power is to deal essentially with emergencies;

(2) Police officers should exercise special care in such cases. Only in exceptional circumstances will it be justifiable to enter into private premises in the context of a domestic dispute to deal with or prevent a further breach of the peace;

(3) The officer must, before he enters, be satisfied on reasonable grounds that there is a real and imminent risk of a breach of the peace;

(4) Where the original breach has not occurred in the police officer’s presence, the further breach of the peace apprehended must be immediate or about to occur;

(5) The threat to the peace must be sufficiently real and present to justify the extreme step of entering private premises and depriving of his liberty a citizen who is not at the time acting unlawfully.

Of course the occupant can file a complaint against or even sue the police (see e.g. Friswell). Whether or not the police are disciplined or must compensate the complainant depends on what they reasonably believed or apprehended about the circumstances.

Case citation credit to Can I come in? The perils of summary entry to a home, on Police Law Blog.

Many jurisdictions have the concept of emergency or exigent circumstances such that they allow the police to enter premises in scenarios like the one in the question, without having to comply with the rules that pertain to non-emergency or non-exigent circumstances (such as having to obtain a warrant beforehand).

Generally speaking, the less urgent or serious the apparent circumstances, the more the police must do to enter; conversely, the more urgent or serious the circumstances, the less the police must do to enter. Some jurisdictions have frameworks that are essentially like the following, from less urgent/serious to more urgent/serious:

apply for a warrant from a magistrate or judge -> seek authorisation from an officer of particular rank or higher -> seek permission from the occupier -> no need to seek permission

england-and-wales

If the police "apprehend serious bodily injury" (Syed v Director of Public Prosecutions [2010] EWHC 81), they can enter the home and they can do so without a warrant. This is provided for at section 17(1)(e) Police and Criminal Evidence Act 1984:

17 Entry for purpose of arrest etc. (1) Subject to the following provisions of this section, and without prejudice to any other enactment, a constable may enter and search any premises for the purpose— ... (e) of saving life or limb or preventing serious damage to property.

At common law, the police also have a power to enter private property without a warrant to stop or prevent a 'breach of the peace'. According to Friswell v Chief Constable of Essex [2004] EWHC 3009 (QB):

(1) The purpose of the power is to deal essentially with emergencies;

(2) Police officers should exercise special care in such cases. Only in exceptional circumstances will it be justifiable to enter into private premises in the context of a domestic dispute to deal with or prevent a further breach of the peace;

(3) The officer must, before he enters, be satisfied on reasonable grounds that there is a real and imminent risk of a breach of the peace;

(4) Where the original breach has not occurred in the police officer’s presence, the further breach of the peace apprehended must be immediate or about to occur;

(5) The threat to the peace must be sufficiently real and present to justify the extreme step of entering private premises and depriving of his liberty a citizen who is not at the time acting unlawfully.

Of course the occupant can file a complaint against or even sue the police (see e.g. Friswell). Whether or not the police are disciplined or must compensate the complainant depends on what they reasonably believed or apprehended about the circumstances.

Case citation credit to Can I come in? The perils of summary entry to a home, on Police Law Blog.